We consider the Cauchy problem for an $n\times n$ strictly hyperbolic system of balance laws $u_t+f(u)_x=g(x,u)$, $x\in\mathbb{R}$, $t>0$, $\|g(x,\cdot)\|_{\mathbf{C}^2}\leq\tilde{M}(x)\in\mathbf{L}^1$, endowed with the initial data $u(0,.)=u_o\in\mathbf{L}^1\cap\mathbf{BV}(\mathbb{R};\mathbb{R}^n)$. Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., $|\lambda_i(u)|\geq c>0$ for all $i\in\{1,\dots,n\}$. Assuming that the $\mathbf{L}^1$ norms of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{\mathbf{BV}(\mathbb{R})}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D.~Amadori, L.~Gosse, and G.~Guerra, {\it Arch.~Ration.~Mech.~Anal.}, 162 (2002), pp.~327--366] to unbounded (in $\mathbf{L}^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.

Guerra, G., Marcellini, F., & Schleper, V. (2009). Balance Laws with Integrable Unbounded Sources. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 41(3), 1164-1189 [10.1137/080735436].

Balance laws with integrable unbounded sources

Abstract

We consider the Cauchy problem for an $n\times n$ strictly hyperbolic system of balance laws $u_t+f(u)_x=g(x,u)$, $x\in\mathbb{R}$, $t>0$, $\|g(x,\cdot)\|_{\mathbf{C}^2}\leq\tilde{M}(x)\in\mathbf{L}^1$, endowed with the initial data $u(0,.)=u_o\in\mathbf{L}^1\cap\mathbf{BV}(\mathbb{R};\mathbb{R}^n)$. Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., $|\lambda_i(u)|\geq c>0$ for all $i\in\{1,\dots,n\}$. Assuming that the $\mathbf{L}^1$ norms of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{\mathbf{BV}(\mathbb{R})}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D.~Amadori, L.~Gosse, and G.~Guerra, {\it Arch.~Ration.~Mech.~Anal.}, 162 (2002), pp.~327--366] to unbounded (in $\mathbf{L}^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.
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hyperbolic balance laws, unbounded sources, pipes with discontinuous cross sections
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Guerra, G., Marcellini, F., & Schleper, V. (2009). Balance Laws with Integrable Unbounded Sources. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 41(3), 1164-1189 [10.1137/080735436].
Guerra, G; Marcellini, F; Schleper, V
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10281/7386